For any set \(A\subset X\) , denote by \(\mathfrak {M}^+_A\)  the cone of all \(\mu \in \mathfrak {M}^+\)  concentrated on
\(A\) , which means that \(A^c:=X\setminus A\)  is locally \(\mu \) -negligible, or equivalently that \(A\)  is \(\mu \) -measurable and \(\mu =\mu |_A\) , \(\mu |_A:=1_A\cdot \mu \)  being the trace of \(\mu \)  to \(A\)  [1]}. (Note that for \(\mu \in \mathfrak {M}^+_A\) , the indicator function \(1_A\)  of \(A\)  is locally \(\mu \) -integrable.) The total mass of \(\mu \in \mathfrak {M}^+_A\)  is \(\mu (X)=\mu _*(A)\) , \(\mu _*(A)\)  and \(\mu ^*(A)\)  denoting the inner and the outer measure of \(A\) , respectively. If moreover \(A\)  is closed, or if \(A^c\)  is contained in a countable union of sets \(Q_j\)  with \(\mu ^*(Q_j)<\infty \) ,If the latter holds, \(A^c\)  is said to be \(\mu \) -\(\sigma \) -finite [1]}. This in particular occurs if the measure \(\mu \)  is bounded (that is, with \(\mu (X)<\infty \) ), or if the locally compact space \(X\)  is \(\sigma \) -compact (that is, representable as a countable union of compact sets [3]}). then for any \(\mu \in \mathfrak {M}^+_A\) , \(A^c\)  is \(\mu \) -negligible, that is, \(\mu ^*(A^c)=0\) . In particular, if \(A\)  is closed, \(\mathfrak {M}^+_A\)  consists of all \(\mu \in \mathfrak {M}^+\)  with support \(S(\mu )\subset A\) , cf. [4]}.
